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2021 | OriginalPaper | Chapter

Computing Rational Points on Rank 0 Genus 3 Hyperelliptic Curves

Authors : María Inés de Frutos-Fernández, Sachi Hashimoto

Published in: Arithmetic Geometry, Number Theory, and Computation

Publisher: Springer International Publishing

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Abstract

We compute rational points on genus 3 odd degree hyperelliptic curves C over \(\mathbb {Q}\) that have Jacobians of Mordell–Weil rank 0. The computation applies the Chabauty–Coleman method to find the zero set of a certain system of p-adic integrals, which is known to be finite and include the set of rational points \(C(\mathbb {Q})\). We implemented an algorithm in Sage to carry out the Chabauty–Coleman method on a database of 5870 curves.
Footnotes
1
The global height is the absolute logarithmic height of the point, which is the maximum of the absolute logarithmic heights of its coordinates. For a rational point \(\frac {n}{d}\), this height is computed as \(\max (\log (|n|),\log (|d|)).\)
 
Literature
2.
go back to reference J. S. Balakrishnan, F. Bianchi, V. Cantoral-Farfán, M. Çiperiani, and A. Etropolski, “Chabauty-Coleman experiments for genus 3 hyperelliptic curves,” in Research Directions in Number Theory, ser. Association for Women in Mathematics Series. Springer, 2019, vol. 19, pp. 67–90. CrossRef J. S. Balakrishnan, F. Bianchi, V. Cantoral-Farfán, M. Çiperiani, and A. Etropolski, “Chabauty-Coleman experiments for genus 3 hyperelliptic curves,” in Research Directions in Number Theory, ser. Association for Women in Mathematics Series. Springer, 2019, vol. 19, pp. 67–90. CrossRef
3.
go back to reference J. S. Balakrishnan, R. W. Bradshaw, and K. S. Kedlaya, “Explicit Coleman integration for hyperelliptic curves,” in Algorithmic Number Theory. ANTS 2010, Lecture Notes in Computer Science. Springer, Berlin, 2010, vol. 6197, pp. 16–31. J. S. Balakrishnan, R. W. Bradshaw, and K. S. Kedlaya, “Explicit Coleman integration for hyperelliptic curves,” in Algorithmic Number Theory. ANTS 2010, Lecture Notes in Computer Science. Springer, Berlin, 2010, vol. 6197, pp. 16–31.
4.
go back to reference A. R. Booker, J. Sijsling, A. V. Sutherland, J. Voight, and D. Yasaki, “A database of genus-2 curves over the rational numbers,” LMS J. Comput. Math., vol. 19, no. suppl. A, pp. 235–254, 2016. A. R. Booker, J. Sijsling, A. V. Sutherland, J. Voight, and D. Yasaki, “A database of genus-2 curves over the rational numbers,” LMS J. Comput. Math., vol. 19, no. suppl. A, pp. 235–254, 2016.
5.
go back to reference W. Bosma, J. Cannon, and C. Playoust, “The Magma Algebra System. I. The User Language,” J. Symbolic Comput., vol. 24, no. 3-4, pp. 235–265, 1997. MathSciNetCrossRef W. Bosma, J. Cannon, and C. Playoust, “The Magma Algebra System. I. The User Language,” J. Symbolic Comput., vol. 24, no. 3-4, pp. 235–265, 1997. MathSciNetCrossRef
6.
go back to reference C. Chabauty, “Sur les points rationnels des courbes algébriques de genre supérieur à l’unité,” C. R. Acad. Sci. Paris, vol. 212, pp. 882–885, 1941. MathSciNetMATH C. Chabauty, “Sur les points rationnels des courbes algébriques de genre supérieur à l’unité,” C. R. Acad. Sci. Paris, vol. 212, pp. 882–885, 1941. MathSciNetMATH
8.
go back to reference ——, “Torsion points on curves and p-adic abelian integrals,” Ann. of Math. (2), vol. 121, no. 1, pp. 111–168, 1985. MathSciNetCrossRef ——, “Torsion points on curves and p-adic abelian integrals,” Ann. of Math. (2), vol. 121, no. 1, pp. 111–168, 1985. MathSciNetCrossRef
10.
go back to reference G. Faltings, “Endlichkeitssätze für abelsche Varietäten über Zahlkörpern,” Invent. Math., vol. 73, no. 3, pp. 349–366, 1983. MathSciNetCrossRef G. Faltings, “Endlichkeitssätze für abelsche Varietäten über Zahlkörpern,” Invent. Math., vol. 73, no. 3, pp. 349–366, 1983. MathSciNetCrossRef
11.
go back to reference W. McCallum and B. Poonen, “The method of Chabauty and Coleman,” Explicit Methods in Number Theory, Rational Points and Diophantine Equations, Panor. Synthèses, vol. 36, pp. 99–117, 2012. MathSciNetMATH W. McCallum and B. Poonen, “The method of Chabauty and Coleman,” Explicit Methods in Number Theory, Rational Points and Diophantine Equations, Panor. Synthèses, vol. 36, pp. 99–117, 2012. MathSciNetMATH
12.
go back to reference J. S. Milne, “Jacobian varieties,” in Arithmetic geometry (Storrs, Conn., 1984). Springer, New York, 1986, pp. 167–212. J. S. Milne, “Jacobian varieties,” in Arithmetic geometry (Storrs, Conn., 1984). Springer, New York, 1986, pp. 167–212.
13.
go back to reference M. Stoll, “On the height constant for curves of genus two. II,” Acta Arith., vol. 104, no. 2, pp. 165–182, 2002. MathSciNetCrossRef M. Stoll, “On the height constant for curves of genus two. II,” Acta Arith., vol. 104, no. 2, pp. 165–182, 2002. MathSciNetCrossRef
14.
go back to reference ——, “Independence of rational points on twists of a given curve,” Compos. Math., vol. 142, no. 5, pp. 1201–1214, 2006. MathSciNetCrossRef ——, “Independence of rational points on twists of a given curve,” Compos. Math., vol. 142, no. 5, pp. 1201–1214, 2006. MathSciNetCrossRef
17.
go back to reference J. L. Wetherell, Bounding the Number of Rational Points on Certain Curves of High Rank, 1997, Ph.D. thesis, University of California at Berkeley. J. L. Wetherell, Bounding the Number of Rational Points on Certain Curves of High Rank, 1997, Ph.D. thesis, University of California at Berkeley.
Metadata
Title
Computing Rational Points on Rank 0 Genus 3 Hyperelliptic Curves
Authors
María Inés de Frutos-Fernández
Sachi Hashimoto
Copyright Year
2021
DOI
https://doi.org/10.1007/978-3-030-80914-0_14

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